Deduction of the formulas used in the ratio calculation

  Some details on the calculation of the Proton/Electron mass ratio

    Thomas Lockyer's model, rudimentary according to his own words, considers that the proton is made up of several nested layers of increasing energies, and that its mass can be calculated from the electric potential energy between these levels.

    My first reaction was disbelief and the suspicion that in the calculation we provided in advance and indirectly the mass of the proton. As his book included the listing of his program in Basic I undertook to translate it into javascript, a very practical language for tinkering and testing a calculation algorithm. And I was able to see that nothing in the calculation appeared ad hoc to obtain this surprising result.

    The calculation is done by starting from the mass of the electron, then adding increasing energies and the total sum of these energies gives exactly the desired ratio. We cannot refine the precision by playing on the number of levels. Let's add or remove a nesting level and we get to a 30% error...!

Let's see the basics of its calculation:

An electron has mass $m_e$, spin angular momentum $\dfrac{\hbar}{2}$, electrical charge $e$, magnetic moment $\mu_e$. The Planck's constant is $h = m_e.c^2.\lambda_e / c$

We have $e = \sqrt{\dfrac{2.\alpha.h}{\mu_0.c}}$ 

with  Permeability of vacuum = $\mu_0 = 4\pi.10^-7$  H/m  and $\alpha$ = Fine-structure constant.

Spin angular momentum is physically equivalent of  'energy-mass' on the end of a radius arm $R_m$ with a certain velocity of giration :

$\dfrac{1}{2}.\hbar = m_e.c.R_m$

Let  $\lambda'_e = \hbar / m_e.c$   then $R_m = \lambda'_e/2$

We use the Bohr Magneton $\mu_B$ to set the idealised charge radius of giration $R_c$

$\mu_B = \dfrac{1}{2}.(e.c.\lambda'_e)$          A.m^2

The model provides two current loops on two parallel planes, per level, and an idealized radius of gyration is assumed : $R_c = \sqrt{2}.R_m = \lambda_e'/\sqrt{2}$

The magnetic moment of the electron ($\mu_e$) differs slightly from the expected value based on its pure spin ($\mu_B$). We will treat this difference as a protrusion $\alpha_u$ in the radius of gyration. 

$$\underbrace{\mu_e = \mu_B.(1 + \alpha_e)    ,  R' = R_c.(1 + \alpha_u/2)}$$

$$\mu_c = 1/2 (e.c.\lambda'_e).(1 + \alpha_e) \equiv 1/2(e.c.\lambda'_e).(1 + \alpha_u/2)^2$$

$$\Longrightarrow  1 + \alpha_e = (1 - \alpha_u/2)^2   \Longrightarrow  \sqrt{1 + \alpha_e} = 1 + \alpha_u/2 \Longrightarrow    \alpha_u = 2.(\sqrt{1 + \alpha_e} - 1)$$

Obtaining normalized lengths for iterative calculation:

  $L_n = \lambda_e'. [(\dfrac{\sqrt{2}}{2})^n  . (1-\sqrt{2}.\alpha_u)]$  

Calculation of the standardized spacing between each level:

$1 = \sqrt{2}/2 + 2d + 2.(/sqrt{2}/2)$  $\Longrightarrow$  $d =  \dfrac{(1 - \sqrt{2}/2)}{2} - \dfrac{\sqrt{2}.\alpha_u}{2}$

 $d$ = corrected distance

The electric potential energy between two charges $q1$ and $q2$ is equal to

$$U = \dfrac{q1.q2}{4\pi.e_0.d} = \dfrac {e^2}{4\pi.e_0.d}$$

with $d$ the distance separating the charges and $e_0$ the permitivity of the vacuum.

Finally we obtain the formula 

$$\triangle_E = 1 + [ \dfrac{e^2}{m_e.c^2.4\pi.\epsilon_0 . \lambda_e' .  (\dfrac{(1 - \sqrt{2}/2)}{2} - \dfrac{\sqrt{2}.\alpha_u}{2})} ]$$

The scale factor $S$ is equal to $1/2d$ :

$$S = \dfrac{1}{[\sqrt{2}/2]^n [1 - \sqrt{2}.\alpha_u]}  \Longrightarrow   m_p / m_e = \triangle_0 + \sum_{n=1}^{18} S . \triangle_n$$

The formula for the Total is therefore:

$$M_p =  \sum_{k=1}^n  \dfrac{e^2}{4\pi.c^2.M_e.e_0.d . [1 - (\sqrt{2}/2 - \sqrt{2}.\alpha) / 2]^n }$$

with $M_e$ = mass of the electron

This is the formula used by the script I put in the first page.

For the details of his theory you can buy his book entitled $Vector Particle Physics$ by T.N.Lockyer Edition of 1992, published with own funds. I had the chance to be able to buy it directly from him by sending a 5 dollar bill by mail (no internet at that time!).